If there exists a real root w then the imaginary part of the left hand side must be 0 for z = w.
- 6i sqrt(3) w2 - 3.i sqrt(3) w = 0.
This is equivalent to 2 w2 + w = 0. So, w = 0 or w = -1/2
But w = 0 can not be a solution of the given equation.
If there exists a real root w then it is -1/2.
Now we check if -1/2 is a root of the given equation and we see that it is so.

The image point of z = a + bi in the Gauss-plane is p.
We rotate p about o and the angle of the rotation is pi/3.
The new position of p is p'.
Calculate the coordinates of p'.

Say z has polar notation r(cos(t)+i sin(t))
p' is the image point of z.(cos(pi/3) + i sin(pi/3)) in the Gauss-plane
Hence, p' is the image point of r(cos(t+pi/3) + i sin(t+pi/3))
The coordinates of p' are (r.cos(t+pi/3) ; r.sin(t+pi/3))

a, b, c are real numbers in the polynomial
p(z) = 2 z4 + a z3 + b z2 + c z + 3 .
Find a such that the numbers 2 and i are roots of p(z) = 0.

Since all the coefficients of p(z) are real, -i is a root of p(z) = 0.
Let 2, i, -i, w be all the roots.
The sum of the roots = 2 + w = -a/2.
The product of the roots = 2w = 3/2.
From this we find w = 3/4 and a = -11/2.

Given:
n is a positive integer.
z is a complex number with modulus 1, such that z2n is not -1.
zn
Show that -------- is a real number
1 + z2n

Calculate all integers n such that zn = (1 + i sqrt(3))n is a real number.

z1 = (1 + i sqrt(3)) has modulus 2 and argument = pi/3.
Thus, zn has modulus 2n and argument n.pi/3.
zn is real if and only if the argument is k.pi (with k = integer).
So, zn is real if and only if n is a multiple of 3.

Calculate the real values of x and y such that (x + iy)3 is real and |x + i y| is higher than 8.

In the Gauss-plane the roots are the vertices of a regular octagon.
The question to resolve here is: give the powers of b that generate all the vertices.
They are the powers br such that r and 8 are coprime.
The eight roots are generated by b, b3, b5, b7.
So, we find 4 values of a namely a1=b ; a2=b3 a3=b5 and a4=b7.

De equation has the form z3=c. The three roots are in the Gauss-plane
the vertices of an equilateral triangle. The three roots have 2 as modulus and the arguments
of z2 and z3 arise by increasing the argument of z1 by 2pi/3 and 4pi/3.